Calculus Examples

$lim_{x \to 0} \frac{x^{2}}{2 \ln (\sec (x))}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} x^{2}}{lim_{x \to 0} 2 \ln (\sec (x))}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} 2 \ln (\sec (x))}$

Step 1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2}}{lim_{x \to 0} 2 \ln (\sec (x))}$

Step 1.2.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{lim_{x \to 0} 2 \ln (\sec (x))}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{2 lim_{x \to 0} \ln (\sec (x))}$

Step 1.3.1.2

Move the limit inside the logarithm.

$\frac{0}{2 \ln (lim_{x \to 0} \sec (x))}$

Step 1.3.1.3

Move the limit inside the trig function because secant is continuous.

$\frac{0}{2 \ln (\sec (lim_{x \to 0} x))}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{2 \ln (\sec (0))}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

The exact value of $\sec (0)$ is $1$ .

$\frac{0}{2 \ln (1)}$

Step 1.3.3.2

The natural logarithm of $1$ is $0$ .

$\frac{0}{2 \cdot 0}$

Step 1.3.3.3

Multiply $2$ by $0$ .

$\frac{0}{0}$

Step 1.3.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{x^{2}}{2 \ln (\sec (x))} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [2 \ln (\sec (x))]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x^{2}]}{\frac{d}{d x} [2 \ln (\sec (x))]}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{2 x}{\frac{d}{d x} [2 \ln (\sec (x))]}$

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2 \ln (\sec (x))$ with respect to $x$ is $2 \frac{d}{d x} [\ln (\sec (x))]$ .

$lim_{x \to 0} \frac{2 x}{2 \frac{d}{d x} [\ln (\sec (x))]}$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \sec (x)$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$lim_{x \to 0} \frac{2 x}{2 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x)])}$

Step 3.4.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to 0} \frac{2 x}{2 (\frac{1}{u} \frac{d}{d x} [\sec (x)])}$

Step 3.4.3

Replace all occurrences of $u$ with $\sec (x)$ .

$lim_{x \to 0} \frac{2 x}{2 (\frac{1}{\sec (x)} \frac{d}{d x} [\sec (x)])}$

Step 3.5

Rewrite $\sec (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{2 x}{2 (\frac{1}{\frac{1}{\cos (x)}} \frac{d}{d x} [\sec (x)])}$

Step 3.6

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$lim_{x \to 0} \frac{2 x}{2 (1 \cos (x) \frac{d}{d x} [\sec (x)])}$

Step 3.7

Multiply $\cos (x)$ by $1$ .

$lim_{x \to 0} \frac{2 x}{2 (\cos (x) \frac{d}{d x} [\sec (x)])}$

Step 3.8

Remove parentheses.

$lim_{x \to 0} \frac{2 x}{2 \cos (x) \frac{d}{d x} [\sec (x)]}$

Step 3.9

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$lim_{x \to 0} \frac{2 x}{2 \cos (x) (\sec (x) \tan (x))}$

Step 3.10

Remove parentheses.

$lim_{x \to 0} \frac{2 x}{2 \cos (x) \sec (x) \tan (x)}$

Step 3.11

Simplify.

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Step 3.11.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.11.1.1

Add parentheses.

$lim_{x \to 0} \frac{2 x}{2 (\cos (x) \sec (x)) \tan (x)}$

Step 3.11.1.2

Reorder $\cos (x)$ and $\sec (x)$ .

$lim_{x \to 0} \frac{2 x}{2 (\sec (x) \cos (x)) \tan (x)}$

Step 3.11.1.3

Rewrite $2 \cos (x) \sec (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{2 x}{2 (\frac{1}{\cos (x)} \cos (x)) \tan (x)}$

Step 3.11.1.4

Cancel the common factors.

$lim_{x \to 0} \frac{2 x}{2 \cdot 1 \tan (x)}$

Step 3.11.2

Multiply $2$ by $1$ .

$lim_{x \to 0} \frac{2 x}{2 \tan (x)}$

Step 3.11.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{2 x}{2 \frac{\sin (x)}{\cos (x)}}$

Step 3.11.4

Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

$lim_{x \to 0} \frac{2 x}{\frac{2 \sin (x)}{\cos (x)}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 2 x \frac{\cos (x)}{2 \sin (x)}$

Step 5

Combine factors.

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Step 5.1

Combine $2$ and $\frac{\cos (x)}{2 \sin (x)}$ .

$lim_{x \to 0} x \frac{2 \cos (x)}{2 \sin (x)}$

Step 5.2

Combine $x$ and $\frac{2 \cos (x)}{2 \sin (x)}$ .

$lim_{x \to 0} \frac{x (2 \cos (x))}{2 \sin (x)}$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Cancel the common factor.

$lim_{x \to 0} \frac{x (2 \cos (x))}{2 \sin (x)}$

Step 6.2

Rewrite the expression.

$lim_{x \to 0} \frac{x (\cos (x))}{\sin (x)}$

Step 7

Convert from $\frac{x (\cos (x))}{\sin (x)}$ to $x \cot (x)$ .

$lim_{x \to 0} x \cot (x)$

Step 8

Consider the left sided limit.

$lim_{x \to 0^{-}} x \cot (x)$

Step 9

Make a table to show the behavior of the function $x \cot (x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & x \cot (x) \\ - 0.1 & 0.99666444 \\ - 0.01 & 0.99996666 \\ - 0.001 & 0.99999966 \end{array}$

Step 10

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $x \cot (x)$ as $x$ approaches $0$ from the left is $1$ .

$1$

Step 11

Consider the right sided limit.

$lim_{x \to 0^{+}} x \cot (x)$

Step 12

Make a table to show the behavior of the function $x \cot (x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & x \cot (x) \\ 0.1 & 0.99666444 \\ 0.01 & 0.99996666 \\ 0.001 & 0.99999966 \end{array}$

Step 13

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $x \cot (x)$ as $x$ approaches $0$ from the right is $1$ .

$1$